I have the following system of ODEs.
\begin{equation} \dot{x} = -x + 4y \end{equation} \begin{equation} \dot{y} = -x - y^3 \end{equation}
Typically, I have seen the first standard guess to be $V(x) = x^2 + y^2$. However, my lecturer opted to use $$V(x) = x^2 + 4 \cdot y^2$$ I am unsure why or how he knew to chose that particular one which is slightly different to the standard guess. Any help would be much appreciated.
Let $a,b$ positive real numbers. Multiplying the first equation by $2ax$, the second one by $2by$, and summing altogether we get $$2ax\dot{x}+2by\dot{y} = -2ax^2 + 8axy -2byx - 2by^4$$ that is $$\dot{V}(x,y)=-2(ax^2 +by^4)+2(4a-b)xy.$$ where $V(x,y)=ax^2+by^2$.
Notice that the $xy$ term, which changes sign in a neighborhood of the origin, disappears as soon as $4a=b$ and we obtain that for any $(x,y)\not=(0,0)$, $$\dot{V}(x,y)=-2a(x^2 +4y^4)< 0$$ with $V(x,y)=a(x^2+4y^2)$.